Clinical statistics for non-statisticians: Day three

Steve Simon

Re-introduce yourself

Here’s one more interesting number about myself

  • 20: I have a 20 year old son.

Tell us one more interesting number about yourself.

Outline of the three day course

  • Day one: Numerical summaries and data visualization
  • Day two: Hypothesis testing and sampling
  • Day three: Statistical tests to compare treatment to a control and regression models

My goal: help you to become a better consumer of statistics

Day three topics

  • Statistical tests to compare a treatment to a control
    • What tests should you use for categorical outcomes?
    • What tests should you use for continuous outcomes?
    • When should you use nonparametric tests?

Day three topics (continued)

  • Regression models
    • How does a regression model quantify trends
    • How does logistic regression differ from linear regression
    • What is a confounding variable
    • How should you control for or adjust for confounding

Figure 1: Image of a passenger jet with four engines

Figure 2: Image of a passenger jet with one bad engine

Figure 3: Image of a passenger jet with two bad engines

Figure 4: Image of a passenger jet with three bad engines

Comparison of treatment and control

  • Treatment, something new to help a patient
    • Active intervention
    • Randomized trial
  • Exposure, something that a patient endures
    • Passive observation
    • Epidemiology study
  • Control
    • Placebo, or
    • Usual standard of care

Comparison of a binary outcome

\(X^2 = \Sigma \frac{(O_{ij}-E_{ij})^2}{E_{ij}}\)

Figure 5: Counts of dead and survived by sex with expected counts

Alternative approach, the odds ratio

        Died  Survived
Females  154       308
Males    709       142

Survival odds for Females 2 to 1 in favor (308 / 154).
Survival odds for Males 5 to 1 against (142/ 709).

Odds ratio = (2/1) / (1/5) = 10

95% CI (7.7, 13)

Alternative approach, relative risk

        Died  Survived
Females  154       308
Males    709       142

Survival probability for 66.7%.
Survival probability for Males 16.7%.

Relative risk = 0.667 / 0.167 = 4

95% CI (3.4, 4.7)

Which is the better measure?

  • Two schools of thought
    • Relative risk is better
      • More natural interpretation
    • Odds ratio is better - Symmetric with respect to outcome
  • Cannot use relative risk for certain datasets

Both are inferior to absolute risk reduction

        Died  Survived
Females  154       308
Males    709       142

Survival probability for 66.7%.
Survival probability for Males 16.7%.

Absolute risk reduction = 0.667 - 0.167 = 0.5

95% CI (0.45, 0.55)

Comparison of multinomial outcome

  • Multinomial = 3 or more categories
  • Beyond the scope of this class
    • Multinomial logistic regression
    • Ordinal logistic regression

Comparison of a continuous outcome

  • Two cases
    • Independent (unpaired) samples
    • Paired samples

Two sample test

  • Is \((\bar{X_1}-\bar{X_2})\) close to zero?
  • How much sampling error?
    • \(S_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\)

Comparison of ages of deaths/survivors

Mean ages 95% CI (-0.3, 3.8)

Paired samples

Room Before  After
 121   11.8   10.1
 125    7.1    3.8
 163    8.2    7.2
 218   10.1   10.5
 233   10.8    8.3
 264   14     12  
 324   14.6   12.1
 325   14     13.7

Average change

Room Before  After Change
 121   11.8   10.1   -1.7
 125    7.1    3.8   -3.3
 163    8.2    7.2   -1.0
 218   10.1   10.5    0.4
 233   10.8    8.3   -2.5
 264   14     12     -2.0
 324   14.6   12.1   -2.5
 325   14     13.7   -0.3  

\(\bar{D}=-1.61,\ S_D=1.24\)

95% CI (-2.65, -0.58)

Assumptions for t-tests

  • t-tests require two or more assumptions
    • Patients are independent
    • Outcome is normally distributed
    • For two sample t-test, equal variation

Nonparametric test

  • Uses ranks of the data
  • Does not rely on normality assumption
  • Does not rely on Central Limit Theorem

Wilcoxon signed rank test

                           Absolute
Room Before  After Change    Change   Rank
 121   11.8   10.1   -1.7       1.7      4
 125    7.1    3.8   -3.3       3.3      8
 163    8.2    7.2   -1.0       1.0      3
 218   10.1   10.5    0.4       0.4      2
 233   10.8    8.3   -2.5       2.5      6/7
 264   14     12     -2.0       2.0      5
 324   14.6   12.1   -2.5       2.5      6/7
 325   14     13.7   -0.3       0.3      1

p = 0.023

Criticisms of nonparametric tests

  • Not easy to get confidence intervals
  • Difficult to do risk adjustments

Figure 6: Quote from “Peggy Sue Got Married

Pop quiz

  • From high school algebra.
    • Pythagorean theorem
      • ?
    • Quadratic formula
      • ?
    • Equation for a straight line
      • ?

Pop quiz answers

  • From high school algebra.
    • Pythagorean theorem
      • \(a^2+b^2=c^2\)
    • Quadratic formula
      • \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
    • Formula for a straight line
      • \(y=mx+b\)

Equation of a straight line

  • \(y=mx+b\)
    • \(m=\) slope \(=\triangle y / \triangle x\)
    • \(b=\) y-intercept

In linear regression

  • y: dependent variable
  • x: independent variable
  • Slope: estimated average change in y when x increases by one unit.
  • Intercept: estimated average value of y when x equals zero.

Example: does mother’s age affect duration of breast feeding?

  • Study of breast feeding with pre-term infants
    • Difficulty: mother leaves hospital first

Figure 7: Scatterplot with regression line for age

Figure 8: Linear regression output

Figure 9: Linear regression output, slope

  • Slope = 0.4
    • The estimated average duration of breast feeding increases by 0.4 weeks for every increase of one year in the mother’s age.

Figure 10: Linear regression output, intercept

  • Intercept = 5.9
    • The estimated average duration of breast feeding is 5.9 weeks for a mother with age = 0.
    • Clearly an inappropriate extrapolation

Figure 11: Linear regression output, p-value

  • p-value=0.019
    • Reject the null hypothesis and conclude that there is a positive relationship between mother’s age and duration of breast feeding.

Figure 12: Linear regression output, confidence interval

  • 95% Confidence interval (0.066 to 0.71)
    • Note 6.626E-02 \(= 6.626 \times 10^{-2}\)
    • You are 95% confident that the true regression slope is positive.

Example: does treatment affect duration of breast feeding?

  • Both groups: encourage breast feeding when mom is in hospital
    • Intervention: feed infants through ng tube when mom is away
    • Control: Feeding using bottles when mom is away

Figure 13: Scatterplot with regression line for treatment=1

Figure 14: Linear regression output, treatment

Figure 15: Scatterplot with regression line for control=1

Linear regression with two independent variables

  • Intercept
  • Slope for first independent variable
  • Slope for second independent variable

Interpretation of intercept and slopes

  • Intercept: estimated average value of y when x1 and x2 both equal zero.
  • Slope for x1: estimated average change in y when x increases by one unit and x2 is held constant.
  • Slope for x2: similar interpretation

Adjusting for covariate imbalance

  • Covariate: variable not of direct interest in the research
    • but has to be accounted for to draw valid conclusions
  • Covariate imbalance: a difference in average levels of the covariate between treatment and control
    • Threat to the validity of the research
  • Example: average age of mothers
    • 25 in control group, 29 in treatment group
  • Covariate imbalance not quite same as confounding

Figure 16: Multiple linear regression output

Adjusted means

  • Unadjusted
    • Treatment: \(12.961 + 0.249 \times 29 = 20.2\)
    • Control: \(12.961 -5.972 + 0.249 \times 25 = 13.2\)
  • Adjusted
    • Treatment: \(12.961 + 0.249 \times 27 = 19.7\)
    • Control: \(12.961 -5.972 + 0.249 \times 27 = 13.7\)

Small group exercises

  • Group 1: Effect of sex and height on fev
  • Group 2: Effect of smoking and age on fev Examples in the medical literature

Joke about prediction models

  • Risks during surgery
    • P[death] = 0.6
  • If risk doubles
    • P[death] = 1.2

Logistic regression

  • Binary outcome
  • Linear on a log odds scale

Figure 17: A linear trend in probability

prob BF = 4 + 2*GA

Figure 18: A bad linear trend in probability

Figure 19: Multiplicative trend in probabilities

Using odds

  • Three to one in favor of victory
    • Expect three wins for every loss
  • Four to one odds against victory
    • Expect four losses for every win
  • Odds = Prob / (1- Prob)
  • Prob = Odds / (Odds + 1)

Odds for winning election to U.S. president in 2024

  • Biden: \(\frac{8/13}{1 + 8/13} = \frac{8}{21} = 0.381\)
  • Trump: \(\frac{1/3}{1 + 1/3} = \frac{1}{4} = 0.25\)
  • DeSantis: \(\frac{1/16}{1+1/16} = \frac{1}{17} = 0.059\)

Probability of winning 2022 World Cup

Brazil: 30.8%
Argentina: 18.2%
France: 16.7%
Spain: 13.3%
England: `10%
Portugal: 7.7%
Netherlands: 5.3%
Croatia: 2.8%

Argentina: \(\frac{0.182}{1-0.182} = 0.2225 \approx 2/9\)

France: \(\frac{0.167}{1-0.167} = 0.2004 \approx 1/5\)

Switzerland: 1.5%
Japan: 1.5%
Morocco: 1.2%
USA: 1.1%
Senegal: 1%
South Korea: 0.67%
Poland: 0.55%
Australia: 0.5%

Odds against winning 2022 football World Cup

Brazil: 9 to 4
Argentina: 9 to 2
France: 5 to 1
Spain: 13 to 2
England: 9 to 1
Portugal: 12 to 1
Netherlands: 18 to 1
Croatia: 35 to 1
Switzerland: 65 to 1
Japan: 65 to 1
Morocco: 80 to 1
USA: 90 to 1
Senegal: 100 to 1
South Korea: 150 to 1
Poland: 180 to 1
Australia: 200 to 1

Figure 20: Multiplicative trend for odds

Additive trend in log odds

Figure 21: Odds converted into probabilities

S-shaped curve

Figure 22: Actual data on gestational age

\(log\ odds = -16.72 + 0.577 \times ga\)

Figure 23: Predicted log odds

  • Let’s examine these calculations for GA = 30.
    • log odds = -16.72 + 0.577*30 = 0.59
    • odds = exp(0.59) = 1.80
    • prob = 1.80 / (1+1.80) = 0.643

Ratio of successive odds

1.01/0.57 = 1.78

1.80/1.01 = 1.78

3.20/1.80 = 1.78

5.70/3.20 = 1.78

Figure 24: Titanic probabilities for death and survival

Figure 25: Logistic regression for Titanic data

  • Female
    • log odds = 0.693
    • odds = 2
    • prob = 0.667

Probability calculations for males

  • Male
    • log odds = 0.693 - 2.301 = -1.608
    • odds = 0.2003
    • prob = 0.167

  • log odds ratio = -2.301
    • odds ratio = 0.1

Figure 26: Description of the interview invite dataset

Figure 27: Crosstabulation of class and interview

Figure 28: Crosstabulation of gender and interview

Figure 29: Logistic regression model for class

Figure 30: Logistic regression model for gender

Figure 31: Fruit fly data, round 1

Figure 32: Fruit fly graph, round 1

Fruit fly data (round 2)

Figure 33: Fruit fly data, round 2

Figure 34: Fruit fly graph, round 2

Figure 35: Fruit fly data, round 3

Figure 36: Fruit fly graph, round 3

Figure 37: Fruit fly graph, estimating the median

Figure 38: Fruit fly graph, estimating survival probability

Figure 39: Hypothetical survival distribution

Figure 40: Hypothetical survival distribution, probability for 20-30 years

Figure 41: Hypothetical survival distribution, probability for 40-60 years

Defining the hazard function (1/2)

  • To make a fair comparison
    • Adjust by the probability of surviving up to age 20 or age 40.
    • Calculate a death rate by dividing by the time range.
    • Calculate over a narrow time interval, Δt.

Defining the hazard function (2/2)

  • The hazard function is defined as
    • h(t) = (P[t ≤ T ≤ t+Δt] / Δt) / P[T ≥ t]
  • Key points
    • adjusted for the number surviving to that time (P[T ≥ t]),
    • calculated as a rate
      • (P[t ≤ T ≤ T+Δt] / Δt) is not a probability, and
    • computed over a narrow time interval.

Figure 42: Increasing, decreasing, and constant hazard functions

Summary

  • Day one: Numerical summaries and data visualization
  • Day two: Hypothesis testing and sampling
  • Day three: Statistical tests to compare treatment to a control and regression models

My goal: help you to become a better consumer of statistics

Any questions?